Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 5, pp. 1205-1226.

In this paper, we will study the existence and qualitative property of standing waves ψ(x,t)=e -iEt ϵ u(x) for the nonlinear Schrödinger equation iϵψ t+ϵ 2 2mΔ x ψ-(V(x)+E)ψ+K(x)|ψ| p-1 ψ=0, (t,x) + × N . Let G(x)=[V(x)] p+1 p-1-N 2 [K(x)] -2 p-1 and suppose that G(x) has k local minimum points. Then, for any l{1,,k}, we prove the existence of the standing waves in H 1 ( N ) having exactly l local maximum points which concentrate near l local minimum points of G(x) respectively as ϵ0. The potentials V(x) and K(x) are allowed to be either compactly supported or unbounded at infinity. Therefore, we give a positive answer to a problem proposed by Ambrosetti and Malchiodi (2007) [2].

DOI: 10.1016/j.anihpc.2010.05.003
Classification: 35J20,  35J60
Keywords: Bound state, Multi-peak solutions, Nonlinear Schrödinger equation, Potentials compactly supported or unbounded at infinity
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     title = {Multi-peak bound states for {Schr\"odinger} equations with compactly supported or unbounded potentials},
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Ba, Na; Deng, Yinbin; Peng, Shuangjie. Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 5, pp. 1205-1226. doi : 10.1016/j.anihpc.2010.05.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.05.003/

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